Philosophy 240: Symbolic Logic Hamilton College Fall 2014 Russell Marcus
Reference Sheeet for What Follows
Names of Languages
PL: Propositional Logic M: Monadic (First-Order) Predicate Logic F: Full (First-Order) Predicate Logic FF: Full (First-Order) Predicate Logic with functors S: Second-Order Predicate Logic
Basic Truth Tables
- á á @ â á w â á e â á / â 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0
Rules of Inference
Modus Ponens (MP) Conjunction (Conj) á e â á á / â â / á A â
Modus Tollens (MT) Addition (Add) á e â á / á w â -â / -á Simplification (Simp) Disjunctive Syllogism (DS) á A â / á á w â -á / â Constructive Dilemma (CD) (á e â) Hypothetical Syllogism (HS) (ã e ä) á e â á w ã / â w ä â e ã / á e ã Philosophy 240: Symbolic Logic, Prof. Marcus; Reference Sheet for What Follows, page 2
Rules of Equivalence
DeMorgan’s Laws (DM) Contraposition (Cont) -(á A â) W -á w -â á e â W -â e -á -(á w â) W -á A -â Material Implication (Impl) Association (Assoc) á e â W -á w â á w (â w ã) W (á w â) w ã á A (â A ã) W (á A â) A ã Material Equivalence (Equiv) á / â W (á e â) A (â e á) Distribution (Dist) á / â W (á A â) w (-á A -â) á A (â w ã) W (á A â) w (á A ã) á w (â A ã) W (á w â) A (á w ã) Exportation (Exp) á e (â e ã) W (á A â) e ã Commutativity (Com) á w â W â w á Tautology (Taut) á A â W â A á á W á A á á W á w á Double Negation (DN) á W --á
Six Derived Rules for the Biconditional
Rules of Inference Rules of Equivalence
Biconditional Modus Ponens (BMP) Biconditional DeMorgan’s Law (BDM) á / â -(á / â) W -á / â á / â
Biconditional Modus Tollens (BMT) Biconditional Commutativity (BCom) á / â á / â W â / á -á / -â
Biconditional Hypothetical Syllogism (BHS) Biconditional Contraposition (BCont) á / â á / â W -á / -â â / ã / á / ã Philosophy 240: Symbolic Logic, Prof. Marcus; Reference Sheet for What Follows, page 3
Rules for Quantifier Instantiation and Generalization
Universal Instantiation (UI)
(á)öá for any variable á, any formula öcontaining á, and öâ any singular term â
Universal Generalization (UG)
öâ for any variable â, any formula öcontaining â, and (á)öá for any variable á
Never UG within the scope of an assumption for conditional or indirect proof on a variable that is free in the first line of the assumption. Never UG on a variable when there is a constant present, and the variable was free when the constant was introduced.
Existential Generalization (EG)
öâ for any singular term â, any formula ö containing â, and (á)öá for any variable á
Existential Instantiation (EI)
(á)öá for any variable á, any formula ö containing á, and öâ any new constant â
Quantifier Equivalence (QE)
(á)öá W -(á)-öá (á)öá W -(á)-öá (á)-öá W -(á)öá (á)-öá W -(á)öá Philosophy 240: Symbolic Logic, Prof. Marcus; Reference Sheet for What Follows, page 4
Rules of Passage
For all variables á and all formulas à and Ä:
RP1: (á)(Ã w Ä) W (á)Ã w (á)Ä RP2: (á)(Ã C Ä) W (á)Ã C (á)Ä
For all variables á, all formulas à containing á, and all formulas Ä not containing á:
RP3: (á)(Ä C Ãá) W Ä C (á)Ãá RP4: (á)(Ä C Ãá) W Ä C (á)Ãá RP5: (á)(Ä w Ãá) W Ä w (á)Ãá RP6: (á)(Ä w Ãá) W Ä w (á)Ãá RP7: (á)(Ä e Ãá) W Ä e (á)Ãá RP8: (á)(Ä e Ãá) W Ä e (á)Ãá RP9: (á)(Ãá e Ä) W (á)Ãá e Ä RP10: (á)(Ãá e Ä) W (á)Ãá e Ä
Here are versions of each of the rules that are less-meta-linguistic and maybe easier to read:
RP1: (x)(Px w Qx) W (x)Px w (x)Qx RP2: (x)(Px C Qx) W (x)Px C (x)Qx RP3: (x)(öC Px) W ö C (x)Px RP4: (x)(ö C Px) W ö C (x)Px RP5: (x)(ö w Px) W ö w (x)Px RP6: (x)(ö w Px) W ö w (x)Px RP7: (x)(ö e Px) W ö e (x)Px RP8: (x)(ö e Px) W ö e (x)Px RP9: (x)(Px e ö) W (x)Px e ö RP10: (x)(Px e ö) W (x)Px e ö
Rules Governing the Identity Predicate (ID)
IDr. Reflexivity: á=á
IDs. Symmetry: á=â W â=á
IDi. Indiscernibility of Identicals
öá á=â / öâ